L(s) = 1 | + 0.838·2-s − 1.90·3-s − 1.29·4-s − 1.59·6-s + 2.46·7-s − 2.76·8-s + 0.624·9-s − 1.49·11-s + 2.46·12-s + 0.929·13-s + 2.07·14-s + 0.274·16-s + 2.80·17-s + 0.523·18-s − 0.0373·19-s − 4.69·21-s − 1.25·22-s + 6.79·23-s + 5.26·24-s + 0.779·26-s + 4.52·27-s − 3.20·28-s + 29-s + 2.15·31-s + 5.75·32-s + 2.83·33-s + 2.35·34-s + ⋯ |
L(s) = 1 | + 0.593·2-s − 1.09·3-s − 0.648·4-s − 0.651·6-s + 0.932·7-s − 0.977·8-s + 0.208·9-s − 0.449·11-s + 0.712·12-s + 0.257·13-s + 0.553·14-s + 0.0685·16-s + 0.680·17-s + 0.123·18-s − 0.00856·19-s − 1.02·21-s − 0.266·22-s + 1.41·23-s + 1.07·24-s + 0.152·26-s + 0.870·27-s − 0.604·28-s + 0.185·29-s + 0.387·31-s + 1.01·32-s + 0.494·33-s + 0.403·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.172757503\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.172757503\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 - 0.838T + 2T^{2} \) |
| 3 | \( 1 + 1.90T + 3T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 + 1.49T + 11T^{2} \) |
| 13 | \( 1 - 0.929T + 13T^{2} \) |
| 17 | \( 1 - 2.80T + 17T^{2} \) |
| 19 | \( 1 + 0.0373T + 19T^{2} \) |
| 23 | \( 1 - 6.79T + 23T^{2} \) |
| 31 | \( 1 - 2.15T + 31T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 + 1.97T + 41T^{2} \) |
| 43 | \( 1 - 9.61T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 + 1.33T + 53T^{2} \) |
| 59 | \( 1 - 4.57T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 0.291T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 0.0141T + 73T^{2} \) |
| 79 | \( 1 - 5.37T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 6.88T + 89T^{2} \) |
| 97 | \( 1 + 9.42T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.66696258098826244770868708955, −9.638017117603174069008407444903, −8.630619342375930429639731218711, −7.83674342936570741763131267267, −6.57828960525751493514817360399, −5.59234191893633496251940922343, −5.10095575703647561672068857856, −4.28109437103757683976855199198, −2.91866736985357939698572894560, −0.909742725494966627195753621109,
0.909742725494966627195753621109, 2.91866736985357939698572894560, 4.28109437103757683976855199198, 5.10095575703647561672068857856, 5.59234191893633496251940922343, 6.57828960525751493514817360399, 7.83674342936570741763131267267, 8.630619342375930429639731218711, 9.638017117603174069008407444903, 10.66696258098826244770868708955